Eventos Anais de eventos
DINAME 2017
XVII International Symposium on Dynamic Problems of Mechanics
Uncertainty analysis of rotating systems
Submission Author:
Helio Fiori de Castro , SP , Brazil
Co-Authors:
Helio Fiori de Castro, Jose Maria Campos dos Santos, Rubens Sampaio
Presenter: Helio Fiori de Castro
doi://10.26678/ABCM.DINAME2017.DIN17-0031
Abstract
Rotordynamics is important in industry, due to the considerable amount of applications of rotating machineries as in energy generation, power transition and others. Because of that, it is very important to understand the dynamical behavior of these machines. Therefore, mathematical models have been developed, considering the rotor, bearings, and supporting structures. Discrete and continuous models are used for the rotor, while the bearings can be represented by concentrated stiffness and damping coefficients. If hydrodynamic journal bearings are considered, these coefficients are obtained from the Reynolds equation, which modeled hydrodynamic pressure distribution of the lubricated oil. Due to the complexity of the models, one should not expect to find analytical solutions for the dynamic behavior of rotating machineries. The simplest of these models of a rotor, named as Jeffcott rotor, considers a massless shaft with a rigid disc fixed in the center of the shaft. In this case, the shaft is supported by rigid bearings. There are others formulations based on the Jeffcott rotor that take into account a non-central, or outboard disc. The main advantage of these models is its simplicity and utility to exhibit the main characteristics of the dynamics. Clearly, a more rigorous quantitative analysis demands more elaborated models. An important aspect that is not traditionally considered in rotordynamics analysis is the robustness of the results. Due to the uncertainties of the properties and conditions of operation, stochastic models must be considered to measure the robustness of the results. The objective of this work is to models the critical speed of a rotating system as random variables taking into account stochastic aspects of rotor dynamics. It is then proposed a stochastic analysis of rotating systems with two bearings and an outboard disc. Campbell diagram is taken into account for this models, because, in this case, there are gyroscopic effects and forward and backward natural frequencies are presented in the response. Since the subject is rather new, the choice of a simple model is justified since our goal is to show the importance of the stochastic aspects. More complex models lead to a significantly higher computational cost. In future works, it is intended to apply this analysis to these more complex models in order to have a more precise analysis. In the analysis, the Young’s modulus and the disc mass are the considered as stochastics. In both case, the standard deviation is 5% of the mean. Gamma distribution is assumed to model these parameters. Monte Carlo simulations are done to quantify the stochastic response of the system. Analyses are carried out for all models considered. For the cases that gyroscopic effect is present, the uncertainty of the forward and backward natural frequency is also obtained. Figure 1 shows the Campbell diagram, considering stochastic parameters. The black line in Figure 1 represents the synchronous excitation. The dark and light green are the respectively first backward and forward natural frequency. As well as the second backward and forward natural frequency are represented by the yellow and blue curves. When the rotating speed is equal to the natural frequency, a synchronous excitation leads to a resonance motion. This rotating speed is called critical speed. Due to the stochastic aspect of the analysis, the critical speeds are represented by statistical distribution, as show in Figure 2. It is important to highlight the significantly variation of the three critical speed. The assumption of a fix operational speed should consider this stochastic analysis, due to the possibility that a small variation in the parameters can bring to a resonance state.
Keywords
Rotordynamics, Uncertainty Analysis., Campbell Diagram
